Finite Math Examples

$0 = 16.3 + 5.69 \cdot t - 0.5 \cdot 9.81 \cdot t^{2}$

Step 1

Rewrite the equation as $16.3 + 5.69 \cdot t - 0.5 \cdot 9.81 \cdot t^{2} = 0$ .

$16.3 + 5.69 \cdot t - 0.5 \cdot 9.81 \cdot t^{2} = 0$

Step 2

Multiply $- 0.5$ by $9.81$ .

$16.3 + 5.69 t - 4.905 t^{2} = 0$

Step 3

Factor $- 0.005$ out of $16.3 + 5.69 t - 4.905 t^{2}$ .

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Step 3.1

Reorder the expression.

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Step 3.1.1

Move $16.3$ .

$5.69 t - 4.905 t^{2} + 16.3 = 0$

Step 3.1.2

Reorder $5.69 t$ and $- 4.905 t^{2}$ .

$- 4.905 t^{2} + 5.69 t + 16.3 = 0$

Step 3.2

Factor $- 0.005$ out of $- 4.905 t^{2}$ .

$- 0.005 (981 t^{2}) + 5.69 t + 16.3 = 0$

Step 3.3

Factor $- 0.005$ out of $5.69 t$ .

$- 0.005 (981 t^{2}) - 0.005 (- 1138 t) + 16.3 = 0$

Step 3.4

Factor $- 0.005$ out of $16.3$ .

$- 0.005 (981 t^{2}) - 0.005 (- 1138 t) - 0.005 \cdot - 3260 = 0$

Step 3.5

Factor $- 0.005$ out of $- 0.005 (981 t^{2}) - 0.005 (- 1138 t)$ .

$- 0.005 (981 t^{2} - 1138 t) - 0.005 \cdot - 3260 = 0$

Step 3.6

Factor $- 0.005$ out of $- 0.005 (981 t^{2} - 1138 t) - 0.005 (- 3260)$ .

$- 0.005 (981 t^{2} - 1138 t - 3260) = 0$

Step 4

Divide each term in $- 0.005 (981 t^{2} - 1138 t - 3260) = 0$ by $- 0.005$ and simplify.

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Step 4.1

Divide each term in $- 0.005 (981 t^{2} - 1138 t - 3260) = 0$ by $- 0.005$ .

$\frac{- 0.005 (981 t^{2} - 1138 t - 3260)}{- 0.005} = \frac{0}{- 0.005}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 0.005$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 0.005 (981 t^{2} - 1138 t - 3260)}{- 0.005} = \frac{0}{- 0.005}$

Step 4.2.1.2

Divide $981 t^{2} - 1138 t - 3260$ by $1$ .

$981 t^{2} - 1138 t - 3260 = \frac{0}{- 0.005}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $- 0.005$ .

$981 t^{2} - 1138 t - 3260 = 0$

Step 5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6

Substitute the values $a = 981$ , $b = - 1138$ , and $c = - 3260$ into the quadratic formula and solve for $t$ .

$\frac{1138 \pm \sqrt{{(- 1138)}^{2} - 4 \cdot (981 \cdot - 3260)}}{2 \cdot 981}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 1138$ to the power of $2$ .

$t = \frac{1138 \pm \sqrt{1295044 - 4 \cdot 981 \cdot - 3260}}{2 \cdot 981}$

Step 7.1.2

Multiply $- 4 \cdot 981 \cdot - 3260$ .

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Step 7.1.2.1

Multiply $- 4$ by $981$ .

$t = \frac{1138 \pm \sqrt{1295044 - 3924 \cdot - 3260}}{2 \cdot 981}$

Step 7.1.2.2

Multiply $- 3924$ by $- 3260$ .

$t = \frac{1138 \pm \sqrt{1295044 + 12792240}}{2 \cdot 981}$

Step 7.1.3

Add $1295044$ and $12792240$ .

$t = \frac{1138 \pm \sqrt{14087284}}{2 \cdot 981}$

Step 7.1.4

Rewrite $14087284$ as $2^{2} \cdot 3521821$ .

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Step 7.1.4.1

Factor $4$ out of $14087284$ .

$t = \frac{1138 \pm \sqrt{4 (3521821)}}{2 \cdot 981}$

Step 7.1.4.2

Rewrite $4$ as $2^{2}$ .

$t = \frac{1138 \pm \sqrt{2^{2} \cdot 3521821}}{2 \cdot 981}$

Step 7.1.5

Pull terms out from under the radical.

$t = \frac{1138 \pm 2 \sqrt{3521821}}{2 \cdot 981}$

Step 7.2

Multiply $2$ by $981$ .

$t = \frac{1138 \pm 2 \sqrt{3521821}}{1962}$

Step 7.3

Simplify $\frac{1138 \pm 2 \sqrt{3521821}}{1962}$ .

$t = \frac{569 \pm \sqrt{3521821}}{981}$

Step 8

The final answer is the combination of both solutions.

$t = \frac{569 + \sqrt{3521821}}{981}, \frac{569 - \sqrt{3521821}}{981}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$t = \frac{569 + \sqrt{3521821}}{981}, \frac{569 - \sqrt{3521821}}{981}$

Decimal Form:

$t = 2.49301889 \dots, - 1.33297812 \dots$